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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spectral analysis of finite convolution operators


Author: Richard Frankfurt
Journal: Trans. Amer. Math. Soc. 214 (1975), 279-301
MSC: Primary 47G05; Secondary 44A35
MathSciNet review: 0397481
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Abstract: In this paper the similarity problem for operators of the form $ ( \ast )\;T:f(x) \to \smallint _0^xk(x - t)f(t)dt$ on $ {L^2}(0,1)$ is studied. Let $ K(z) = \smallint _0^1\;k(t){e^{itz}}dt$. A function $ C(z)$ is called a symbol for T if $ C(z)$ can be written in the form $ C(z) = K(z) + {e^{iz}}G(z)$, where $ G(z)$ is a function bounded and analytic in a half plane $ y > \delta $, for some real number $ \delta $. Under suitable restrictions, it is shown that two operators of the form $ ( \ast )$ will be similar if they possess symbols which are asymptotically close together as $ z \to \infty $ in some half plane $ y > \delta $.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0397481-9
PII: S 0002-9947(1975)0397481-9
Article copyright: © Copyright 1975 American Mathematical Society